This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel 1/(1 − θx) in place of the celebrated exponential kernel exp(θx) of the exponential families theory. We extend the theory to cover generating measures with support that is unbounded on one side. We illustrate the need for such an extension by showing that cubic… (More)
In this paper, we give three equivalent properties of the class of multivariate simple cubic natural exponential families (NEF's). The first property says that the cu-mulant function of any basis of the family is a solution of some Monge-Ampère equation, the second is that the variance function satisfies a differential equation, and the third is… (More)
The Wishart distribution on an homogeneous cone is a generalization of the Riesz distribution on a symmetric cone which corresponds to a given graph. The paper extends to this distribution, the famous Olkin and Rubin characterization of the ordinary Wishart distribution on symmetric matrices.
We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when the domain of means can be extended beyond the " natural domain " .
For a natural exponential family (NEF), one can associate in a natural way two standard families of conjugate priors, one on the natural parameter and the other on the mean parameter. These families of conjugate priors have been used to establish some remarkable properties and characterization results of the quadratic NEF's. In the present paper, we show… (More)
The aim of this paper is to study the mixture of the Riesz distribution on symmetric matrices with respect to the multivariate Poisson distribution. We show, in particular, that this distribution is related to the modified Bessel function of the first kind. We also study the generated natural exponential family. We determine the domain of the means and the… (More)
In this paper, we introduce, for a multiplier χ, a notion of generalized power function x → ∆ χ (x), defined on the homogeneous cone P of a Vinberg algebra A. We then extend to A the famous Gindikin result, that is we determine the set of multipliers χ such that the map θ → ∆ χ (θ −1), defined on P * , is the Laplace transform of a positive measure R χ. We… (More)
Bobecka and Wesolowski (2002) have shown that, in the Olkin and Rubin characterization of the Wishart distribution (See Casalis and Letac (1996)), when we use the division algorithm defined by the quadratic representation and replace the property of invariance by the existence of twice differentiable densities, we still have a characterization of the… (More)